SPHERICAL-ABERRATION-FREE UNSTABLE RESONATOR WITH SPHERICAL PRIMARY MIRROR

HAL Id: jpa-00220607

https://hal.archives-ouvertes.fr/jpa-00220607

Submitted on 1 Jan 1980

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

SPHERICAL-ABERRATION-FREE UNSTABLE

RESONATOR WITH SPHERICAL PRIMARY

MIRROR

J. Perkins, C. Cason

To cite this version:

JOURNAL

DE

PHYSIQUE Colloque C9, suppZIment a u n O 1 l , Tome 41, novembre 1980, page C9-393

SPHERICAL-ABERRATION-FREE UNSTABLE RESONATOR W I T H S P H E R I C A L PRIMARY MIRROR

J.F.

Perkins and c.casoni.

.University o f AZabama i n HuntsviZZe, H u n t s v i l l e , Alabama 35804, U.S.A.

A m i X r e c t e d Energy D i r e c t o r a t e , U.S. Army M i s s i l e ~onunand, Redstone A r s e n a l , AZabama 35809, U.S. A.

RQsum6.- Les rCsonateurs instables conventionnels pour lasers utilisent des surfaces sphdriques pour les deux miroirs. L'on a montrd que des problsmes de qualit6 optiques dues aux deviations se produi- sent pour les longueurs d'onde courtes et/ou pour les grandes dimensions de biais. Ce papier ddter- mine la portde des paramstres au del5 desquels les miroirs sphdriques ne peuvent plus Ctre utilisds de fason satisfaisante et propose un projet modifis qui r6tablit la qualitd optique essentiellement utilis6e.

Abstract.- Conven~ional unstable resonators for lasers use spherical surfaces for both mirrors. Op-

--

tical quality problems due to aberrations are shown to arise for short wavelengths and/or large cross dimensions. This paper determines the range of parameters beyond which spherical mirrors may not be satisfactorily used, and proposes a modified design which restores the essentially idealized optical quality.

INTRODUCTION

Unstable resonators have been widely and suc- cessfully used for high-energy lasers, and have been extensively described in the literature. [1 - 51 A special advantage of such resonators is the absence of an inherent limitation to the spatial width of the fundamental transverse mode; as a result, the gain region does not necessarily need to be made very narrow as compared to its length as is the case

for stable resonators. The half-width,

A,

of the output beam, the mirror spacing,

L,

and the wave- length,

A ,

can be combined into the tube Fresnel

2

number, F = A /(XL) which serves as a convenient T

parameter for discussion. While the mode-volume- filling properties readily permit operation for indefinitely large values of FT, the relative dim- ensions for many cases of interest have been such

2

that F is moderately small, perhaps less than 10

.

T

Restriction to such a range of values of FT has various ramifications. For relatively small values of FT, diffraction effects are substantial; hence detailed optical modelling of transverse modes in

this regime requires wave-optics methods. Extensive wave-optics calculations have been carried out by various groups [I - 51; results include interesting quasi-periodicity of empty-resonator mode loss as a function of Fresnel number [I, 3

-

51 and misalign- ment properties. [ 4

-

51 (The asymptotic-expansion methods of Horwitz [ 3 -

41

are not necessarily lim- ited to such relatively small values of FT.) For this range of parameters, resonator mirrors with spherical figures perform satisfactorily so far as optical aberrations are concerned, and most papers on unstable-resonator calculations seem at least tacitly to assume that the mirrors are spherical in shape.

For substantially larger values of F the situ- T

ation is different. Wave-optics calculations would be costly in computer storage and time because of the very large number of mesh points required to represent diffractive structure in detail. Presum- ably, however, wave-optics calculations are not really necessary in this regime; even for empty resonators, the diffractive structure evidently be-

C9-394 JOURNAL DE PHYSIQUE

comes s m a l l i n a m p l i t u d e . [ 3 1 A s a n a l t e r n a t i v e one may c o n s i d e r i t e r a t i v e r a y - o p t i c s c a l c u l a t i o n s ; we have r e c e n t l y w r i t t e n such a computer code. Another f e a t u r e of r e s o n a t o r s w i t h v e r y l a r g e v a l u e s of F T is t h a t d e t a i l e d s h a p e s of m i r r o r s u r f a c e s become i m p o r t a n t ; t h i s f e a t u r e forms t h e s u b j e c t of t h e p r e s e n t r e p o r t . I t i s p r e f e r a b l e t o u s e s p h e r i c a l m i r r o r s whenever t h e y p r o v i d e a c c e p t a b l e o p t i c a l q u a l i t y , because of t h e i r g r e a t e r s i m p l i c i t y of p r o d u c t i o n and t e s t i n g a s compared t o more i n v o l v e d m i r r o r s h a p e s s u c h a s v a r i o u s c o n i c s e c t i o n s of r e v o l u t i o n . The purpose of t h e p r e s e n t p a p e r i s t o d e t e r m i n e t h e p a r a m e t e r s p a c e beyond which s p h e r i c a l m i r r o r s may n o t b e s a t i s f a c t o r i l y used i n u n s t a b l e r e s o n a t o r s and t o i n v e s t i g a t e m i r r o r - f i g u r e combi- n a t i o n s which r e s t o r e t h e e s s e n t i a l l y i d e a l i z e d out- p u t o p t i c a l q u a l i t y w h i l e minimizing a d d i t i o n a l complexity and c o s t . MODEL FORMULATI014

The c o n f i g u r a t i o n of primary i n t e r e s t , a s il- l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e l , i s a two-mirror, standing-wave u n s t a b l e r e s o n a t o r of p o s i t i v e - b r a n c h , c o n f o c a l t y p e . The i d e a l i z e d o u t p u t from such a r e s o n a t o r h a s uniform i n t e n s i t y (over t h e unobscured

CONFOCAL POSITIVE-BRANCH UNSTABLE RESONATOR

W L -4

F i g u r e 1. U n s t a b l e Resonator

p a r t of t h e beam) and c o n s t a n t phase. D e v i a t i o n s of t h e o u t p u t wave from i d e a l i t y r e d u c e t h e amount of e n e r g y f a l l i n g w i t h i n t h e c e n t r a l f a r - f i e l d r e g i o n . One c o n v e n i e n t measure of t h e s c a l a r o p t i c a l q u a l i t y (SOQ) i s t h e r a t i o 1/10 of t h e a c t u a l i n t e n s i t y , I , i n t h e c e n t e r of t h e f a r f i e l d , w i t h t h e r e s u l t s of d e v i a t i o n s from i d e a l i t y t a k e n i n t o a c c o u n t , t o t h e c e n t r a l i n t e n s i t y , 10, which would b e a c h i e v a b l e w i t h a n i d e a l beam. The p r e s e n t s t u d y t r e a t s depar- t u r e s from u n i f o r m i t y of p h a s e b e c a u s e t h e SOQ i s l e s s s e n s i t i v e t o d e p a r t u r e s from uniform amplitude. For r e l a t i v e l y s m a l l d e p a r t u r e from t h e d e s i r e d uni- p h a s e p r o p e r t y , t h e r e d u c t i o n i n c e n t r a l f a r - f i e l d b r i g h t n e s s v a r i e s i n a s i m p l e manner w i t h t h e r o o t - mean-square (r.m.s.) d e p a r t u r e of phase from i d e a l - i t y , A$rms. I t h a s been shown [61 t h a t

I n a g e o m e t r i c - o p t i c s approach t h e i n c r e m e n t a l phase A@ i s simply c a l c u l a t e d a s a r a t i o of a n i n c r e m e n t a l v a l u e AZ of t h e t o t a l l e n g t h of a r a y and t h e o p t i c a l wavelength, i . e . Here AZ r e p r e s e n t s t h e o p t i c a l p a t h d i f f e r e n c e , which i s e l s e w h e r e r e f e r r e d t o a s OPD.

The p r e s e n t work combines closed-form approx- i m a t e c a l c u l a t i o n s of e x p e c t e d a b e r r a t i o n s of o u t - p u t from a r e s o n a t o r and d e t a i l e d r e s u l t s of numer- i c a l r a y - t r a c i n g c a l c u l a t i o n s . The p r e s e n t i t e r a - t i v e r a y - t r a c i n g code, IRAYT, i s more g e n e r a l t h a n one used f o r s t u d i e s of t i l t e d - s p h e r i c a l - m i r r o r r e s o n a t o r s . The IRAYT code w i l l be o n l y v e r y b r i e f l y mentioned h e r e . It t r e a t s m i r r o r s w i t h s u r f a c e s which can b e a t l e a s t a s g e n e r a l a s r o t a t e d c o n i c s e c t i o n s . Following Spencer and Murty [ 7 1 ,

t h e s u r f a c e f u n c t i o n of t h e m i r r o r i s t a k e n t o b e

2

F(X,Y,Z) = Z

-

CP

1

+

( 1 - KC2P2)%

Here we have

2 2

p 2 = (X + Y ) .

The v e r t e x c u r v a t u r e i s c. The conic parameter K

determines t h e t y p e of c o n i c s e c t i o n , i n accordance w i t h Table I.

Types of Surfaces Associated w i t h Various Values of K

Range o r Value Type of S u r f a c e of K K < O Hyperboloid K = O Paraboloid O < K < ~ Hemelipsoid of r e v o l u t i o n about major a x i s ~ = l Hemisphere ~ > l Hemelipsoid of r e v o l u t i o n about minor a x i s Table I

RESULTS AND DISCUSSION

An approximate p r e d i c t i o n of t h e amount of s p h e r i c a l a b e r r a t i o n from a r e s o n a t o r w i t h s p h e r i c a l m i r r o r s can be made by c o n s i d e r i n g t h e o p t i c a l p a t h d i f f e r e n c e between d i s t a n c e s along a nominal r a y p a t h i n a s p h e r i c a l m i r r o r system and i n a c o r r e - sponding p a r a b o l o i d a l m i r r o r system ( i . e . , one having t h e same v a l u e s of v e r t e x c u r v a t u r e ) . The o r i g i n of c o o r d i n a t e s h a s been placed a t t h e i n t e r - s e c t i o n of t h e o p t i c a x i s w i t h t h e m i r r o r . The s u r f a c e e q u a t i o n can be w r i t t e n a s

where

The d e s i r e d s o l u t i o n f o r Z i s given by

Now p/R i s s m a l l compared t o u n i t y , and i t i s u s e f u l t o w r i t e t h e f i r s t few terms of t h e s e r i e s expansion of t h e r a d i c a l . This g i v e s

The v a l u e a s s o c i a t e d with r e t a i n i n g only t h e f i r s t two terms of t h e s e r i e s expansion i s t h e v a l u e asso- c i a t e d w i t h a p a r a b o l o i d a l m i r r o r . Thus t h e i n c r e - mental v a l u e o f L i s approximated by t h e t h i r d term

R

i n t h e square b r a c k e t s . The o p t i c a l p a t h d i f f e r e n c e i s double t h i s amount i n magnitude. We can t h e n w r i t e a s t h e lowest-order approximation t o t h e i n c r e m e n t a l OPD a s s o c i a t e d with t h e primary m i r r o r

,.

4

For t h e secondary m i r r o r one has

Because of t h e n a t u r e of a c o n f o c a l r e s o n a t o r t h e r a d i u s of c u r v a t u r e

$

of t h e primary m i r r o r i s M times a s l a r g e i n magnitude a s t h e r a d i u s of curva- t u r e RS of t h e secondary m i r r o r (where M i s t h e r e s o n a t o r m a g n i f i c a t i o n ) . Also, t h e v a l u e of pp a s s o c i a t e d with t h e i n t e r s e c t i o n of t h e r a y w i t h t h e primary m i r r o r w i l l be M times a s l a r g e a s t h e v a l u e of PS a s s o c i a t e d with t h e i n t e r s e c t i o n of t h e r a y w i t h t h e secondary m i r r o r . Thus we can r e w r i t e t h e above a s

The two OPD c o n t r i b u t i o n s tend t o cancel. The over- a l l OPD, A Z , is ( t o lowest o r d e r ) given by

A l l t h e above i s f o r a s i n , g l e round-trip through t h e r e s o n a t o r . Numerical r a y - t r a c i n g c a l - c u l a t i o n s have been performed f o r s e v e r a l c a s e s and compared t o t h e above p r e d i c t i o n s ; t h e agreement was q u i t e s a t i s f a c t o r y .

(3-396 JOURNAL DE PHYSIQUE

SENSITIVITY OF SPHERICAL ABERRATION INDUCED DEGRADATION dependence is On

%'

the radius of curvature Of the

IN BEAM QUALITY FOR SQUARE APERTURES ON GEOMETRIC OUTCOUPLING

p r i m a r y m i r r o r . We c a n r e c a s t t h i s i n t e r m s of t h e m i r r o r s e p a r a t i o n L a s t h e independent v a r i a b l e : F o r t h e s e l f - c o n s i s t e n t m u l t i p l e - p a s s r e s o n a t o r s i t u a t i o n , t h e t o t a l i n c r e m e n t a l OPD due t o s p h e r - i c a l a b e r r a t i o n i s simply t h e sum of c o n t r i b u t i o n s f o r v a r i o u s v a l u e s of pp, where t h e f i r s t s u c h v a l u e "ma,

_,

e t c . The s e r i e s can i s p t h e second is

--

m a ~ ' b e r e a d i l y summed, and g i v e s f o r t h e m u l t i - p a s s i n c r e m e n t : When w r i t t e n i n t e r n s of L a s t h e independent v a r i - a b l e , t h i s becomes The e q u a t i o n s d e r i v e d above p r e d i c t a c o n t r i b u t i o n t o o p t i c a l p a t h d i f f e r e n c e due t o s p h e r i c a l a b e r - r a t i o n which can be w r i t t e n a s 2 2 2 4 A 2 =

c5

(X

+

Y ) = c5p

.

For t h e s q u a r e - a p e r t u r e c a s e , t h e root-mean-square v a l u e of t h e OPD averaged o v e r t h e e n t i r e a p e r t u r e c a n b e shown t o be: 4 RMSOPD = 0.6753 A C5.

The closed-form p r e d i c t i o n f o r C5 i s g i v e n above. T h i s may b e r e c a s t i n t e r m s of t h e t u b e F r e s n e l number FT, where . 2 We t h e n o b t a i n RMSOPD = 0.0211 (1

-

1/M) 4 1 X 2 2 -4

1

F~ ' ( 1 - M ) The p r e d i c t e d dependence of o u t p u t o p t i c a l q u a l i t y on beam-width 2A f o r v a r i o u s v a l u e s of g e o m e t r i c o u t c o u p l i n g i s i l l u s t r a t e d i n F i g u r e 2. The v a l u e s were computed u s i n g t h e closed-form e x p r e s s i o n s g i v e n above, and a few p o i n t s were

F i g u r e 2. Output O p t i c a l Q u a l i t y

produce t h e same amount of beam-quality d e g r a d a t i o n non-vanishing ( i . e . , f o r p a r a b a l o i d a l m i r r o r s ) , a s measured by t h e S t r e h l r a t i o . The dependence o f a l l a b e r r a t i o n s would v a n i s h . We wish t o

t h e RMSOPD on M i s c o n t a i n e d i n two f a c t o r s , t h e choose v a l u e s of t h e c o n i c p a r a m e t e r s K~ and K s u c h P p r i n c i p a l one of which i s t h a t t h e magnitudes of t h e f o u r t h - o r d e r c o n t r i b u t i o n s

(9)

*.

from each of t h e m i r r o r s a r e e q u a l and hence c a n c e l .

I

We must compare t h e terms f o r v a l u e s o f Y which a r e But of c o u r s e t h e dependence of RMSOPD on t h e h a l f -

4 i n t h e r a t i o of M. Thus we r e q u i r e t h a t

w i d t h A i s of t h e form A

.

Thus. t h e v a l u e of beam w i d t h a t which a g i v e n l e v e l of beam-quality degrad-

a t i o n o c c u r s w i l l b e a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e q u a n t i t y M _{T h i s immediately l e a d s t o t h e r e q u i r e m e n t t h a t} (M

-

1 )

-

KS = M Kp. T h i s p r e d i c t i o n i s i n agreement w i t h t h e p l o t s . For i n s t a n c e , t h e p r e d i c t e d beam-width r a t i o f o r 65% When t h e r a t i o of v a l u e s of t h e c o n i c p a r a m e t e r s i s o u t c o u p l i n g a s compared t o 85% o u t c o u p l i n g i s 1.5. M a s r e q u i r e d by t h e above e q u a t i o n , t h e f o u r t h - o r d e r It is c l e a r t h a t f o r r e s o n a t o r s of l a r g e t r a n s - c o n t r i b u t i o n s h o u l d v a n i s h f o r a n i d e a l l y - a l i g n e d s y s - v e r s e s i z e , t h e o p t i c a l q u a l i t y w i l l b e e n t i r e l y tem.

u n a c c e p t a b l e i f b o t h m i r r o r s h a v e a s p h e r i c a l s u r f a c e . The above e q u a t i o n a p p l i e s , of c o u r s e , whenever We r e s t r i c t a t t e n t i o n t o m i r r o r s which a r e c o n i c b o t h m i r r o r s a r e e l l i p s o i d a l , and i n c l u d e s t h e spe- s u r f a c e s of r e v o l u t i o n , and d e r i v e a c o n d i t i o n on c i a l c a s e s of s p h e r i c a l m i r r o r s , p a r a b o l o i d s , and r e l a t i v e v a l u e s of t h e c o n i c p a r a m e t e r s K~ and K

S e l l i p s o i d s . C e r t a i n s p e c i a l c a s e s a r e of p a r t i c u l a r s u c h t h a t t h e s p h e r i c a l a b e r r a t i o n v a n i s h e s . Sub- i n t e r e s t . F i r s t , p a r a b o l o i d a l m i r r o r s o b v i o u s l y s c r i p t s P and S a r e used t o d e n o t e primary and s a t i s f y t h e above r a t i o r e q u i r e m e n t s i n c e b o t h K~

s e c o n d a r y m i r r o r s , r e s p e c t i v e l y , of a n o m i n a l l y con- and Kp a r e z e r o . When t h e p r i m a r y m i r r o r i s s p h e r -

f o c a l u n s t a b l e r e s o n a t o r of m a g n i f i c a t i o n M. We i c a l , i . e . , Kp = 1, t h e s e c o n d a r y ( i . e . , feedback)

c o n s i d e r t h e s u r f a c e e q u a t i o n , g i v e n above, f o r a m i r r o r w i l l need t o b e a s u r f a c e of r e v o l u t i o n a b o u t c o n i c s u r f a c e of r e v o l u t i o n . For s i m p l i c i t y one t h e minor a x i s , w i t h K = M. When t h e secondary

S

may s e t X = 0, and r e w r i t e t h e e q u a t i o n a s m i r r o r i s s p h e r i c a l , i . e . , KS = 1, t h e primary m i r - r o r w i l l need t o b e a s u r f a c e of r e v o l u t i o n about t h e major a x i s , w i t h K =

P M '

By expanding i n powers of Y and k e e p i n g o n l y t h e The above closed-form p r e d i c t i o n of c a n c e l l a t i o n f i r s t two t e r m s one o b t a i n s o f s p h e r i c a l a b e r r a t i o n by a p p r o p r i a t e c h o i c e of t h e

The m i r r o r c u r v a t u r e s a r e i n t h e r a t i o

r a t i o of c o n i c p a r a m e t e r s of t h e two m i r r o r s was confirmed by IRAYT c a l c u l a t i o n s . The c a l c u l a t e d RMSOPD was reduced by some t h r e e o r d e r s of magnitude a s compared t o i t s v a l u e f o r a p a i r of s p h e r i c a l m i r r o r s .

JOURNAL DE PHYSIQUE

CONCLUSIONS

The i d e a l l a s e r r e s o n a t o r would u s e c o n f o c a l p a r a b o l o i d a l m i r r o r s and b e f r e e from a l l a b e r r a - t i o n s . U n s t a b l e r e s o n a t o r s w i t h l a r g e t r a n s v e r s e dimensions w i l l produce u n a c c e p t a b l y poor o u t p u t o p t i c a l q u a l i t y i f b o t h m i r r o r s a r e s p h e r i c a l i n s h a p e , due t o s p h e r i c a l a b e r r a t i o n of t h e system. I n c a s e s where t h e primary m i r r o r i s a v a i l a b l e a s a s p h e r i c a l s u r f a c e i t is c l e a r t h a t t h e p r o p e r c h o i c e of a n a s p h e r i c secondary m i r r o r can e l i m i n a t e low o r d e r s p h e r i c a l a b e r r a t i o n . Low o r d e r a b e r r a - t i o n e f f e c t s g r e a t l y r e d u c e t h e beam q u a l i t y f o r l a s e r r e s o n a t o r s having l a r g e FT. T h i s s t u d y h a s shown t h a t t h e a s p h e r i c s u r f a c e s e l e c t e d s h o u l d be an e l l i p s o i d of r e v o l u t i o n i f a c o n i c s e c t i o n s u r - f a c e i s d e s i r e d . It should b e n o t e d t h a t t h i s e l l i p s o i d does n o t d e p a r t g r e a t l y from a s p h e r i c a l s u r f ace. REFERENCES

1. A. E. Siegman, Appl. Opt.

2,

353 (1974). 2. E. A. S z i k l a s and A.E. Siegman, Appl. Opt. 1 4 , 1874 (1975).

-

3 . P. Horwitz, J. Opt. Soc. Am. 63, 1528 (1973).

4. P. Horwitz, Appl. Opt.

15,

167 (1976). 5. J . F. P e r k i n s and C h a r l e s Cason, Appl. Phys. L e t t .

3 ,

198 (1977).

6. M. Born and E. Wolf, P r i n c i p l e s of O p t i c s , 3rd. Ed. (Pergamon P r e s s , New York, 1965).

7. G . H. Spencer and M. V . R. K . Murty, J .